Of
critical importance in the transmission of HIV are "gatekeepers,"
the HIV-negative partners of persons who are HIV-infected.
These are the persons at risk and they are the persons who
can eventually spread the disease further. And since the highest
infectivity comes in the first months after infection, usually
before knowledge of infection, the behavior of these "gatekeepers"
while they are HIV- is critical. In a sample of 267 persons
from high drug use neighborhoods, we collected data on 3254
relationships involving 1271 other persons. We quantitatively
describe the gradient of infection potential by which HIV
can diffuse from the HIV+ population through "gatekeepers"
to the rest of the population through both drug injection
behaviors and sex behaviors.

The
Limits of Sexual Network Data: Implications for Mathematical
Modelling of STISlides:htmlpdfpspptPaper:pdf

Empirical
and theoretical studies have highlighted the importance of
the local (egocentric) and the global (sociocentric) network
structure on the individual risk of infection and the spread
of diseases in populations. Transmission dynamics models of
STI (sexually transmitted infection) and HIV/AIDS have been
instrumental in highlighting the importance of quantifying
sexual behaviour such as the average and variance in sexual
activity, the mixing pattern, concurrency and others in order
to understand epidemiological trends. Detailed individual
based models of partnership formation and dissolution (micro-simulation
or network models) are increasingly being used in order to
capture the full complexity of sexual networks and "more realistically"
simulate the course of STI and HIV/AIDS. However, at the moment,
the complexity of models has outstripped the level of behavioural
data currently available for most populations.

The
objective of the talk is to review and discuss the limits
of currently available sexual network data and the implication
for mathematical modelling of STI. It will be shown how the
limited network data available, compounded by our incomplete
understanding of individual behaviour, has a number of consequences
for the formulation and validation of network models, the
interpretation of model results, and the formulation of research
questions and data collection. In the case of a lethal disease,
like HIV/AIDS, not only is it important to understand the
impact of the network structure on the spread of disease,
but is it also important to understand and assess the impact
of the spread of disease on the network structure. It will
be shown that this is particularly relevant to understand
the different impacts of antiretroviral therapy or vaccination
in heterogeneous populations largely afflicted by the epidemic.

Stephen P. Borgatti (Department of Organization Studies
in the Carroll School of Management, Boston College) borgatts@bc.edu

Issues in Identifying Structurally important Nodes in NetworksSlides:pdfPaper:pdf

This
paper considers the problem of identifying sets of structurally
important nodes in a network. A number of related issues are
considered. First, I outline the different reasons why we might
want to identify key nodes, showing that different measures
(known as centrality measures in the social network literature)
are needed for each. Second, I show how structural importance
is affected by the manner in which things flow through a network.
A typology is constructed to classify types of flows in terms
of the kinds of trajectories that are possible (e.g., paths,
trails or walks) and whether transmission occurs via serial
replication, parallel replication or transfer. The relative
importance of nodes in a given graph changes depending on what
kind of flow process is operating. Third, I consider the problem
of measuring the importance of sets of nodes. Finally, the subtext
of the paper is a commentary on the nature of centrality measures
and what they are intended to measure.

The
amount and geographic patterning of human genetic variation
is an evolutionary consequence of several historical and contemporary
processes including population expansion, demographic subdivision,
and migration. Quantifying this variation, and in turn the extent
to which these forces have acted in shaping human genetic structure
is a key component to understanding the evolution of human populations.
Here I present an analysis framework based upon graph-theory,
which we call Population Graphs (PG). While the PG framework
allows the extraction of traditional population genetic statistics
such as differentiation (ST)
and isolation by distance , topological analysis
of the connectedness among human populations provides heretofore-unattainable
information on intra-population evolutionary history. I highlight
the utility of the PG framework using data consisting of 1056
individuals assayed for 376 variable microsatellite loci sampled
from 52 populations around the globe. An analysis of the topology
of the human graphs reveals the following characteristics of
human population genetic structure. First, groups of populations
exhibit significant topological structuring consistent with
geographically relevant population subdivisions. Second, the
topological distances among populations are significantly correlated
with geographic separation supporting the notion of isolation
by distance and spatially proximate migration patterns. Finally,
we show how the topology of the graph is used to identify specific
populations whose patterns of connectivity prove to be critical
to the movement of genetic information across the entire graph.

Contact tracing, followed by treatment, is a key control measure
in the battle against infectious diseases. It represents an
extreme form of locally targetted control, a hyper-parasite
acting on infection, and as such has the potential to be highly
efficient, especially when dealing with low numbers of cases.
Modelling contact tracing requires explicit information about
the transmission pathways from each individual and hence the
network of contacts. Using pair-wise approximations and full
stochastic simulations to model network-based processes, the
utility of contact tracing is investigated. A simple relationship
between the efficiency of contact tracing necessary to eradicate
infection and te basic reproductive ratio of the disease is
shown to hold in a wide variety of scenarios. Only clustering
within the transmission network is found to destroy this relationship,
enhancing the effectiveness of contact tracing by providing
alternative tracing pathways. Since the critical efficiency
depends on the characteristics of individuals within the network,
applying different tracing regimes within differing subpopulations
can achieve the elimination of infection whilst lowering the
burden on health care services.

We
describe the estimation of very large, realistic social contact
networks and study their structural properties. As a specific
example, we consider the social network for the city of Portland,
Oregon, developed as a part of the TRANSIMS/EpiSims project
at the Los Alamos National Laboratory. The most complete description
of the network is a bipartite graph, with two types of nodes:
people and locations; where edges represent people visiting
locations on a typical day. We describe various computationally
tractable projections and approximations of this network and
compare their structural properties.

Simon
D.W. Frost (Department
of Pathology, University of California, San Diego, Antiviral
Research Center) sdfrost@ucsd.edu

Simulation
of Epidemiological Models on Networks (poster
session)

Joint
work with Klaus G. Muller.

Individual-
or agent-based simulations are useful tools for understanding
how the spread of an infectious agent or computer virus is affected
by the structure of the underlying contact network and by the
natural history of infection at an individual level. Spurred
by the lack of freely available software to simulate epidemic
processes on networks, we are developing a package, Epydemic,
based upon SimPy (http://simpy.sourceforge.net), an open-source
discrete-event simulation library written in Python, that permits
the rapid prototyping of epidemic models. Infections consisting
of multiple stages are easily and concisely modeled using semi-coroutines.
The software includes a graphical user interface for parameter
entry, result visualization etc., and an interactive console
that allows the user to directly analyze components of the simulation.
The poor performance normally associated with the use of an
interpreted language for simulation is compensated for; by the
use of efficient algorithms for the contact process and for
the scheduling of events; by run-time compilation; by the use
of extension modules programmed in C; and by parallelization
of model runs using the Message Passing Interface. We present
an example of a standard SIR model spreading in a configuration
graph.

We
consider statistical and stochastic models for graphs that can
be used to represent the structural characteristics of network.
To date, the use of graph models for networks has been limited
by three interrelated factors: the complexity of realistic models,
paucity of empirically relevant simulation studies, and a poor
understanding of the properties of inferential methods. In this
talk we discuss solutions to these limitations. We emphasize
the important of likelihood-based inferential procedures and
role of Markov Chain Monte Carlo (MCMC) algorithms for simulation
and inference. A primary ongoing issue is the identification
of classes of realistic and parsimonious models. In this regard
show the unsuitability of some commonly promoted Markov models
classes because they can result in degenerate probability distributions.
The ideas are motivated and illustrated by the study of sexual
relations networks with the objective of understanding the social
determinants of HIV spread.

One impediment to the statistical analysis of network
data has been the difficulty in modeling the dependence among
the observations. In the very simple case of binary (0-1) network
data, some researchers have parameterized network dependence
in terms of exponential family representations. Accurate parameter
estimation for such models is difficult, and the most commonly
used models often display a significant lack of fit. Additionally,
such models are generally limited to binary data. In contrast,
mixed effects models have been a widely successful tool in capturing
statistical dependence for a variety of data types, and allow
for prediction, imputation, and hypothesis testing within a
general regression context. We propose novel random effects
structures to capture network dependence, which can also provide
graphical representations of network structure and variability.

There is increasing interest in modeling network data using
exponential random graph models (ERGMs). Fitting these models
using traditional methods such as maximum likelihood is difficult
if not impossible due to the fact that evaluation of the likelihood
function involves a summation with a very large number of terms.
This talk discusses a method that uses stochastic approximation
of the likelihood function based on a Markov chain Monte Carlo
(MCMC) approach. An alternative approach known as maximum pseudolikelihood
is also discussed.

Big
Worlds, Isolated Individuals: Some Characteristics of Social
Networks of Ordinary People

Belief in the idea of a 'small world' as applied to human societies
lacks empirical support. Even the data that originally gave
rise to this idea (Milgram) does not support it. The purpose
here is to consider some characteristics of social networks
of randomly selected (Table of Random Numbers) ordinary urban
residents as a first step away from speculations about networks
in modern societies towards solid empirical evidence. Measures
presented will include graph- theoretic mean and median distances,
and eccentricities. At the core of this work is the presupposition
that to more fully understand factors affecting the spread of
many human pathogens it is necessary to be able to accurately
characterize the underlying population networks through which
they can be transmitted.

Key problems for models of disease spread relate to threshold, velocity
of spread, final size and control. All of these depend crucially
on the network structure of individual interactions.

Networks of interest range from the local extreme where interactions
are only between nearest neighbours in some low dimensional
space, and the infinite-dimensional 'mean-field' extreme where
all interact equally with all. Intermediate cases of practical
interest include 'small-world' and meta-population models.

I shall discuss the various structures of such models, their
similarities and differences, and some approximations to them.
The main aim is to identify what features of contact structure
need to be captured when formulating a model for any specific
problem of disease spread.

Epidemic
Potential in Human Sexual Networks: Connectivity and The Development
of STD CoresSlides:htmlpdfpsppt

Research
on the spread of many sexually transmitted diseases suggests
that infection results from a heterogeneous distribution of
infected actors. While infection rates are too low in the population
at large to account for epidemic spread, within smaller groups
infection rates are high enough to generate outbreaks. This
paper shows that network node connectivity provides a natural
measure for such STD cores. After describing STD cores and node
connectivity, I demonstrate the plausibility of the model with
data from prior epidemic outbreaks, discuss the std-core implications
of popular large-scale network models, and show how STD cores
can develop with small changes in local behavior.

The
Influence of Concurrent Partnerships on Network Structure and
Transmission Dynamics

Concurrent
partnerships are defined as partnerships that overlap in time,
rather than obeying the norm of sequential monogamy. Concurrency
changes the structure of a sexual partnership network, creating
larger components and providing an ecology that favors the rapid
spread of infectious pathogens. Simulation studies have demonstrated
that concurrent partnerships can increase the speed and pervasiveness
of spread in a population, even when the number of partnerships
is held constant. Empirical studies have shown that concurrency
is associated with higher levels of transmission. Thus, there
is growing evidence that concurrency may play an important role
in explaining the differentials in prevalence across population
subgroups. This study uses three nationally representative data
sets to identify the levels and variation of concurrency in
the US, Thai, and Ugandan populations. Both the levels and the
patterns of concurrent partnerships are very different in these
populations. Simulation based on these patterns show transmission
dynamics that replicate the observed variation in prevalence.

Mark
E.J. Newman
(Center for the Study of Complex Systems, University of Michigan)
mejn@umich.edu

I will discuss observational results on the statistical properties
of contact networks, including degree distributions, correlation
properties, transitivity, and assortativity. I will also discuss
models of disease propagation on networks that have these properties
and describe how the properties in question affect the spread
of the disease. Some results are already well known, such as
the fact that highly degree-assortative networks support epidemics
at lower density, but others are not, such as the fact that
high levels of transitivity cause diseases to saturate populations
in regimes only slightly above the epidemic threshold.

Pattison
and Robins (2002) argued that social networks can be modelled
as the outcome of processes that occur in overlapping local
regions of the network, termed local social neighbourhoods.
Within this framework, each neighbourhood is conceived as a
possible site of interaction and is associated with a subset
of possible network ties. Global network structure is then hypothesised
to arise as the outcome of processes occurring within these
overlapping local neighbourhoods. In this paper, we review theoretical
arguments for various hypotheses about the forms of these local
neighbourhoods. We consider Markovian neighbourhoods as well
as generalized realisation-dependent neighbourhoods that
are generated, in part, by interactive network processes themselves.
We introduce some promising new neighbourhood forms including
what we term k-triangles, multiple triadic structures sharing
a common dyadic base. We also introduce hypothesized relationships
between parameters representing related neighbourhood structures,
such as k-stars or k-triangles, and thereby obtain simplified
model specifications. Illustrative empirical analyses based
on these various model specifications are presented.

Exponential
Random Graph (p*) Models for Social Networks: The Global Outcomes
of Local Model Specifications

Exponential
random graph models, when derived from a dependence graph using
the Hammersley-Clifford theorem, are specified in terms of local
network structures. But as these localized patterns agglomerate,
the global outcomes are often not apparent. We review our recent
work on simulating distributions of Markov random graphs, examining
the resulting global structures by comparison with appropriate
Bernoulli distributions of graphs. We provide examples of various
stochastic global "worlds" that may result, including small
worlds, long path worlds and dense non-clustered worlds with
many four-cycles. Degeneracy in these models relates to the
movement from structure to randomness, when parameter scaling
results in a phase transition occurring at a certain "temperature".
Degenerate or "frozen" deterministic structures may be merely
empty or full graphs, but also include more interesting highly
clustered "caveman" graphs, bipartite structures, and global
cyclic structures involving structurally equivalent groups.

But
Markov random graphs are only one possible way to specify exponential
random graphs. We present recent results from simulations for
two other new model specifications. The first specification
includes binary attribute measures on the nodes, with network
ties and actor attributes mutually contingent, resulting in
joint social influence/social selection models. The second specification
includes aggregations of triangle and star counts, permitting
an explicit model form for the degree distribution and a new
transitivity concept, k-triangles, reflecting the distribution
of triangles across the graph.

Richard
Rothenberg, MD (Department of Family and Preventive
Medicine, Emory University School of Medicine) rrothen@emory.edu

Large
Network Concepts and Small Network CharacteristicsSlides:htmlpdfpsppt

The
characteristics of large networks—degree distribution,
small world phenomena, community structure, assortatitivity,
clustering, vulnerability to attack, etc.—may not be fully
recognizable in the smaller (by 5-7 orders of magnitude) networks
within which disease transmission takes place. The smoothing
afforded by size may not protect small network from unpredictable
manifestations that result from underlying heterogeneity and
attendant sampling variability. To explore the characteristics
of small networks, we have assembled 15 data sets from completed
network studies that focused on the transmission of STDs and
HIV. These studies reveal underlying heterogeneity in their
demographic characteristics, risk behaviors, and disease prevalence,
but some similarities with regard to degree distribution, clustering
and, depending on the predominant risks, assortatitivity. For
example, the aggregated degree distribution (a composite totaling
>14,000 dyads) is scale-free (that is, the Cumulative Probability
Distribution is linear in the log-log scale, thus fitting a
power law curve with an coefficient of ~2.0), as are most of
the degree distribution for individual studies (albeit with
considerably more “noise”). Clustering greater than
that predicted for random graphs is present in all the network
studies that permitted examination of components. Short mean
path lengths between persons within components, a manifestation
of the small world effect, are universally present. Such observations
provide substantiation that small networks may behave similarly
to large ones, despite greater variability and sampling uncertainties
and provide some empirical validation of the theoretical basis
for an apparent lack of epidemic threshhold and continued low
level endemic disease transmission of some STDs and HIV in these
microsettings.

The
task of slowing the spread of HIV is complicated by our difficulties
in collecting accurate information about certain key subpopulations,
such as injection drug users and commercial sex workers. Using
a new sampling and estimation method called respondent-driven
sampling, researchers are now able to collect information about
some of these key subpopulations more quickly, cheaply, and
accurately than before.

A
respondent-driven sample is selected with a snowball-type design
(sample members recruit their friends). Despite the numerous
biases inherent in this sample selection process, an estimation
procedure is developed which, under specified (and quite general)
conditions, can be used to make unbiased estimates about the
proportion of the population with a specific trait -- for example
the percentage of injection drug users in a city with HIV. The
estimation procedure uses the sample to make inference about
the social network connecting the subpopulation. This network
information is then used to make inference about the characteristics
of the subpopulation. It is also the case that these estimates
are asymptotically unbiased no matter how the seeds (initial
members of the sample) are selected.

Sacle-free
Networks and Sexually Transmitted Diseases: A Description of
Observed Patterns of Sexual Contacts in Britain and Zimbabwe
(poster session)

Sexually
transmitted infections spread through a network of contacts
created by the formation of sexual partnerships. Methods developed
in physics can characterise a wide range of networks through
a description of the distribution of numbers of sex partnerships.
It has been suggested that in the Swedish population this 'degree'
distribution follows a power law and therefore indicates a 'scale-free'
network. Our objectives were to test statistically whether distributions
of numbers of sexual partners reported by different populations
and over different time periods are well described by power
laws and to estimate their exponent and its implications. Maximum
likelihood estimates of the exponent of a scale free network
fitted to reported distributions of numbers of partners are
compared with the fit for an exponential null model. Data are
taken from 4 population based surveys, three from Britain and
one from rural Zimbabwe. We find that the networks can be described
by a power law over a number of orders of magnitude. In addition,
exponents differ significantly and meaningfully, with an 'accelerating
network' formed between men who have sex with men (MSM). Networks
with an exponent indicating the lack of a 'critical spread rate'
are also found for the other populations except for women in
Britain. Thus statistical analyses demonstrate that a scale-free
network approach provides a reasonable description of distributions
of reported numbers of sexual partners. Further, if these networks
are formed over a short time only a very small transmission
probability will be sufficient to lead to persistence of infection.

Stochastic
Simulation of Epidemics on Large Contact Networks (poster
session)

Joint
work with Martin Eichner (Department
of Medical Biometry, University of Tübingen, Germany).

We
have implemented a fast stochastic individual-based simulator
the analysis of disease transmission and containment interventions.
The simulator consists of a discrete event simulator for the
processing of event-based models and plug-ins for different
contact network topologies.

The
discrete event simulation distinguishes tree types of events.
The first type implements the standard SEIRS infection dynamics
with susceptible, exposed, infectious and recovered states as
well as vaccination and a simple birth/death process. The second
type models the visibility of the disease according to none,
detectable or obvious symptoms. The third type allows to model
intervention strategies (like contact tracing, quarantine and
case isolation), which influence the contact structure of individuals.
Events can trigger further events for the same individual and
via the contact network for other individuals. All events are
processed in a discrete event simulator which is optimized for
large numbers of events using a priority queue (indirect heap
algorithm) and can process about 50.000 events per second.

The
inhabitants of the population are represented by their internal
state (infection, symptom and contact status) and represent
nodes in a contact network. The modular design allows to exchange
the contact network independent of the chosen discrete event
model. For each individual the contact network allows to identify
a limited number of contacts for transmission of the infection
or for implementing contact tracing interventions. Currently
there exist parameterized network generators for local, global,
random and scalefree contact networks. Moreover, the data struc-ture
allows to maintain arbitrary networks consis-ting of several
independent layers. We were able to simulate populations of
two million individuals on a personal computer.

Simulation-Based
Statistical Inference for Evolution of Social NetworksSlides:pdf

Social
networks are structures of social ties between individuals (or
other social actors, e.g., companies or countries). The most
common representation of a social network is a directed graph,
with individuals as nodes, in which the arcs indicate for each
of the ordered pairs of individuals whether the tie in question
(e.g., friendship) is present or not. In addition, there can
be covariate data, referring to the individuals or the dyads
(ordered or unordered pairs of nodes).

Social
mechanisms can be brought to light much better using longitudinal
observations on social networks, than by using single observations.
The most common type of longitudinal observations are repeated
measures, or panel data. Repeated measures on social networks
represent a complicated data structure. Computer simulation
offers fruitful possibilities here, because it greatly expands
the scope of modeling beyond the models for which likelihood
and other functions can be analytically calculated. Continuous-time
models are more appropriate for modeling longitudinal social
network data than discrete-time models because of the endogenous
feedback processes involved in network evolution.

Probability
models for the evolution of social networks are discussed which
are based on the idea of actor-oriented modeling: the nodes
in the network represent actors who change their relations in
a process of optimizing their "objective function", plus a random
component representing unexplained change. The resulting model
constitutes a continuous-time Markov chain, and can be simulated
in a straightforward manner. Similar models can be proposed
which are tie- oriented, in which ties can change randomly as
a result of optimizing an objective function. In all these models,
the change in the network is modeled as the stochastic result
of network effects (reciprocity, transitivity, etc.) and effects
of covariates. The model specification is given by the rate
function, defining the rate at which ties may change; the objective
function, indicating the "preferred" state of the network; and
the gratification function, representing change tendencies for
which the drive for creation of a new tie is not the opposite
of the drive for deletion of this tie when it exists. These
three functions may depend on endogenous and/or exogenous actor
and dyad characteristics.

The
parameters of this model are weights in the rate, objective,
and gratification functions, which may depend on network structure
and on covariates. The parameters can be estimated using a stochastic
version of the method of moments, implemented by a Robbins-Monro-type
algorithm. An example is given of the evolution of the friendship
network in a group of university freshmen students.

Universal Behavior in a Generalized Model of ContagionSlides:htmlpdfpsppt

Models of contagion arise broadly both in the biological and
social sciences, with applications ranging from the transmission
of infectious disease (1, 2) to the diffusion of innovations
(3, 4) and the spread of cultural fads (5-7). We present a general
model of contagion which, by explicitly incorporating memory
of past exposures to, for example, an infectious agent, rumor,
or new product, includes the main features of existing contagion
models as special limiting cases, and interpolates between them.
We study in detail a simple version of the model, finding that
under general conditions only three classes of collective dynamics
exist, two of which correspond to familiar epidemic threshold
(8) and critical mass (9) dynamics, while the third is a distinct
intermediate case. Furthermore, we find that for a given length
of memory, the class into which a particular system falls is
determined entirely by the values of two variables, each of
which ought to be measurable empirically. Our model suggests
novel measures for assessing the susceptibility of a population
to large contagion events, and also a possible strategy for
inhibiting or facilitating them.